SYSTEMS OF LINEAR INEQUALITIES Solving Linear Systems of Inequalities by Graphing

SYSTEMS OF LINEAR INEQUALITIES

Solving Linear Systems of Inequalities by Graphing

Solving Systems of Linear Inequalities

1. We show the solution to a system of linear inequalities by graphing them.

a) This process is easier if we put the inequalities into Slope-Intercept Form, y = mx + b.

Solving Systems of Linear Inequalities

2. Graph the line using the y-intercept & slope.

a) If the inequality is < or >, make the lines dotted.

b) If the inequality is < or >, make the lines solid.

Solving Systems of Linear Inequalities

3. The solution also includes points not on the line, so you need to shade the region of the graph:

a) above the line for ‘y >’ or ‘y ’.b) below the line for ‘y <’ or ‘y ≤’.

Solving Systems of Linear Inequalities

Example: a: 3x + 4y > - 4

b: x + 2y < 2

Put in Slope-Intercept Form: ) 3 4 4

4 3 4

31

4

a x y

y x

y x

) 2 2

2 2

11

2

b x y

y x

y x

Solving Systems of Linear Inequalities

a:

dotted

shade above

b:

dotted

shade below

Graph each line, make dotted or solid and shade the correct area.

Example, continued:

3: 1

4 a y x 1

: 12

b y x

Solving Systems of Linear Inequalities

a: 3x + 4y > - 4

3: 1

4 a y x

Solving Systems of Linear Inequalities

a: 3x + 4y > - 4

b: x + 2y < 2

3: 1

4 a y x

1: 1

2 b y x

Solving Systems of Linear Inequalities

a: 3x + 4y > - 4

b: x + 2y < 2

The area between the green arrows is the region of overlap and thus the solution.

2 2

1

y x

y x

2

2 1

x y

y x

The dry cleaner charges $4 to clean a pair of pants and $3 to clean a shirt. You want to get at least 8 items cleaned. You have $32 to spend on dry cleaning.a. Write a system of inequalities to model the situation.

Suppose you are buying two kinds of notebooks for school. A spiral notebook costs $2, and a three-ring notebook costs $5. You must have at least 6 notebooks. The cost of the notebooks can be no more than $20.a. Write a system of inequalities to model the situation.b. Graph and solve the system.

A camp counselor needs no more than 30 campers to sign up for two mountain hikes. The counselor needs at least 10 campers on the low trail and at least 5 campers on the high trail.a. Write a system of inequalities to model the situation.b. Graph and solve the system.